标题： 最小值和Bethe自由能的临界点在因子图的形变收缩下保持不变 显示英文标题

标题： Minima and Critical Points of the Bethe Free Energy Are Invariant Under Deformation Retractions of Factor Graphs

摘要： 在图模型、因子图以及更一般的基于能量的模型中，变量之间的相互作用由图、超图或在最一般情况下由偏序集（poset）编码。 由于相互作用的基础结构中存在循环，因此无法对这类概率模型进行精确推理。 相反，人们通过优化Bethe自由能来进行近似变分推理。 Bethe自由能的临界点对应于相关信念传播算法的固定点。 对于具有有限变量的一般图、超图和偏序集，这些临界点的完整表征仍然是一个开放问题。 我们表明，对于链长最多为1的超图和偏序集，将概率模型的相互作用偏序集改为具有相同同伦类型的偏序集，会在相关自由能的临界点之间产生双射。 这一结果扩展并统一了经典结果，这些结果假设特定的可收缩形式以证明Bethe自由能临界点的唯一性。 显示英文摘要

摘要： In graphical models, factor graphs, and more generally energy-based models, the interactions between variables are encoded by a graph, a hypergraph, or, in the most general case, a partially ordered set (poset). Inference on such probabilistic models cannot be performed exactly due to cycles in the underlying structures of interaction. Instead, one resorts to approximate variational inference by optimizing the Bethe free energy. Critical points of the Bethe free energy correspond to fixed points of the associated Belief Propagation algorithm. A full characterization of these critical points for general graphs, hypergraphs, and posets with a finite number of variables is still an open problem. We show that, for hypergraphs and posets with chains of length at most 1, changing the poset of interactions of the probabilistic model to one with the same homotopy type induces a bijection between the critical points of the associated free energy. This result extends and unifies classical results that assume specific forms of collapsibility to prove uniqueness of the critical points of the Bethe free energy.